I Haag's Theorem: Explain Free Field Nature

[lindberg]

TL;DR

Can someone explain in simple terms why, according to Haag's theorem, a free field cannot become an interacting one?

What is the main reason for a free field staying free according to Haag's theorem?

 

[Demystifier]

The formal transformation from a free to an interacting field turns out to be mathematically ill defined due to an IR divergence (infinite volume in which the fields live). For details, I highly recommend the book A. Duncan, The Conceptual Framework of Quantum Field Theory, section 10.5 How to stop worrying about Haag's theorem.

 

[PeterDonis]

according to Haag's theorem, a free field cannot become an interacting one?

That's not quite what Haag's theorem says. A free field and an interacting field are different things, and one cannot "become" the other, regardless of what Haag's theorem or any other mathematical result says.

Haag's theorem says, basically, that free fields and interacting fields live in different, unitarily inequivalent Hilbert spaces. To someone who is used to the usual way of modeling interacting fields as perturbations of free fields, this seems like a problem; but there are other approaches to quantum field theory, such as the algebraic approach, in which it is not a problem at all.

 

[lindberg]

To someone who is used to the usual way of modeling interacting fields as perturbations of free fields, this seems like a problem; but there are other approaches to quantum field theory, such as the algebraic approach, in which it is not a problem at all.

Wasn't Haag's conclusion extended later to other approaches?
I might be wrong, don't hesitate to correct me.

An Algebraic Version of Haag’s Theorem​

https://link.springer.com/article/10.1007/s00220-011-1236-7

 

[PeterDonis]

Wasn't Haag's conclusion extended later to other approaches?

Given that the whole point of the algebraic approach to QFT is to be able to deal with unitarily inequivalent representations, showing that the algebraic approach leads to unitarily inequivalent representations isn't much of an issue.